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, Volume 23, Issue 2–3, pp 197–209 | Cite as

A Bipartite Analogue of Dilworth’s Theorem

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Abstract

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is greater than every element of B or every element of A is incomparable with every element of B. We prove that \(m(n)=\Theta\left(\frac{n}{\log n}\right)\). Moreover, for fixed ε ∈ (0,1) and n sufficiently large, we construct a partially ordered set P with n elements such that no element of P is comparable with \(n^{\varepsilon } \) other elements of P and for every two disjoint subsets A and B of P each with cardinality at least \(\frac{14n}{\epsilon\log_2 n}\), there is an element of A that is comparable with an element of B.

Key words

Dilworth’s theorem Ramanujan graph convex compact sets 
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References

  1. 1.
    Alon, N.: Eigenvalues and expanders. Theory of computing (Singer Island, Fla., 1984). Combinatorica 6(2), 83–96 (1986)MATHMathSciNetGoogle Scholar
  2. 2.
    Alon, N.: Ramsey graphs cannot be defined by real polynomials. J. Graph Theory 14(6), 651–661 (1990)MATHMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Milman, V.D.: \(\lambda\sb 1,\) isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory Ser. B 38(1), 73–88 (1985)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Alon, N., Pach, J., Pinchasi, R., Radoičić, R., Sharir, M.: Crossing patterns of semi-algebraic sets. J. Comb. Theory Ser A 111, 310-326 (2005)MATHCrossRefGoogle Scholar
  5. 5.
    Barak, B., Kindler, G., Shaltiel, R., Sudakov, B., Wigderson, A.: Simulating independence: new constructions of condensers, Ramsey graphs, dispersers and extractors. In: Proc. of the 37th ACM STOC, pp. 1–10 (2005)Google Scholar
  6. 6.
    Benczúr, A., András, A., Förster, J., Király, Z.: Dilworth’s theorem and its application for path systems of a cycle — implementation and analysis. Algorithms – ESA ’99 (Prague). Lecture Notes Computer Science, vol. 1643, pp. 498–509. Springer, Berlin Heidelberg New York (1999)Google Scholar
  7. 7.
    Berge, C.: Les problèmes de coloration en théorie des graphes. Publ. Inst. Stat. Univ. Paris 9, 123–160 (1960)MATHMathSciNetGoogle Scholar
  8. 8.
    Chiu, P.: Cubic Ramanujan graphs. Combinatorica 12(3), 275–285 (1992)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229.Google Scholar
  10. 10.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(2), 161–166 (1950)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Erdős, P., Hajnal, A., Pach, J.: Ramsey-type theorem for bipartite graphs. Geombinatorics 10, 64–68 (2000)MathSciNetGoogle Scholar
  12. 12.
    Erdős, P., Komlós, J.: On a problem of Moser. Combinatorial theory and its applications, I. (Proc. Colloq., Balatonfüred, 1969), pp. 365–367. North-Holland, Amsterdam (1970)Google Scholar
  13. 13.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)Google Scholar
  14. 14.
    Fox, J., Pach, J.: A bipartite analogue of Dilworth’s theorem for multiple partial orders, preprint.Google Scholar
  15. 15.
    Gessel, I., Rota, G-C (ed.): Classic Papers in Combinatorics. Birkhauser Boston, MA (1987)MATHGoogle Scholar
  16. 16.
    Graham, R.L., Rothschild, B.L., Spencer, J.: Ramsey Theory, 2nd edn. John Wiley, New York (1990)MATHGoogle Scholar
  17. 17.
    Greene, C., Kleitman, D.J.: The structure of Sperner k-families. J. Comb. Theory Ser. A. 20(1), 41–68 (1976)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Larman, D., Matoušek, J., Pach, J., Töröcsik, J.: A Ramsey-type result for convex sets. Bull. Lond. Math. Soc. 26(2), 132–136 (1994)MATHGoogle Scholar
  19. 19.
    Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Matoušek, J., Welzl, E.: Good splitters for counting points in triangles. J. Algorithms 13(2), 307–319 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Morgenstern, M.: Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q. J. Comb. Theory Ser. B. 62(1), 44–62 (1994)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Murty, M.R.: Ramanujan graphs. J. Ramanujan Math. Soc. 18(1), 1–20 (2003)MATHMathSciNetGoogle Scholar
  23. 23.
    Pach, J., Solymosi, J.: Crossing patterns of segments. J. Comb. Theory Ser. A. 96, 316–325 (2001)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Pach, J., Törőcsik, J.: Some geometric applications of Dilworth’s theorem. Discrete Comput. Geom. 12(1), 1–7 (1994)MATHMathSciNetGoogle Scholar
  25. 25.
    Pach, J., Tóth, G.: Comments on Fox News. Geombinatorics 15, 150–154 (2006)MATHMathSciNetGoogle Scholar
  26. 26.
    Pudlák, P., Rödl, V.: Pseudorandom sets and explicit constructions of Ramsey graphs. Complexity of computations and proofs. Quad. Mat. 13, 327–346, Dept. Math., Seconda Univ. Napoli, Caserta, Italy (2004)MATHGoogle Scholar
  27. 27.
    Seinsche, D.: On a property of the class of n-colorable graphs. J. Comb. Theory Ser. B. 16, 191–193 (1974)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Tietze, H.: Über das Problem der Nachbargeibiete im Raum. Monatshefte Math. 16, 211–216 (1905)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Tóth, G., Valtr, P.: Geometric graphs with few disjoint edges. 14th Annual ACM Symposium on Computational Geometry, Minneapolis, MN, 1998. Discrete Comput. Geom. 22(4), 633–642 (1999)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets. Dimension Theory. Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD 1992Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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